Line Shellings of Geometric Lattices

TL;DR

This paper introduces a new shelling method for the lattice of flats in a matroid using line shellings, providing a novel proof of shellability for these structures and their nested set complexes.

Contribution

It presents the concept of line shellings for matroid lattices and offers a new proof of shellability for the order complex of these lattices.

Findings

Proves shellability of the order complex of matroid lattices.

Establishes shellability for all nested set complexes of matroids.

Introduces line shellings as a new combinatorial tool.

Abstract

Inspired by Bruggesser-Mani's line shellings of polytopes, we introduce line shellings for the lattice of flats of a matroid: given a normal complex for a Bergman fan of a matroid induced by a building set, we show that the lexicographic order of the coordinates of its vertices is a shelling order. This gives a new proof of Bj\"orner's classical result that the order complex of the lattice of flats of a matroid is shellable, and demonstrates shellability for all nested set complexes for matroids.